Hinckley & Bosworth BC hit problems after chief executive Jim Corry left to join Wyre BC earlier this month.
It abandoned an attempt to appoint an acting chief executive and decided instead to seek a secondee.
This left it with no head of paid service, as this post must be held by a member of staff.
Mr Cook said: 'I enjoy a close working relationship with the East Midlands regional Local Government Association, who offer positive support to all authorities in its membership. We are currently working on a number of projects together.'
Ms Claridge said the council suspended the hunt for a temporary chief executive when unions complained that the internal advertisement had excluded the 36 third-tier managers.
'It will take three or four months to fill the post and we cannot have no one in control,' she said.
'We could not take the risk of going ahead with a temporary appointment and being challenged.'
Opposition leader David Bill (Lib Dem) said the council was in 'a most extraordinary situation' and would refer the matter to the Standards Board.